Nonlinear Hilbert Adjoints: Properties and Applications to Hankel Singular Value Analysis

Abstract:
The notion of an adjoint operator for a nonlinear mapping has few interpretations in the literature. Our extension has appeared in a less general form in the nonlinear functional analysis context for characterizing G\^ateaux differentiable homogeneous operators. In systems theory another extension was defined in a purely state space context via an adjoint system based on Hamiltonian extension methods. In this paper, a third extension is proposed, a so called nonlinear Hilbert adjoint. It is shown to unite the two existing concepts and provides an essential tool for singular value analysis at nonlinear Hankel operators.